×
Bagul, Yogesh J.; Chesneau, Christophe; Kostić, Marko

On the Cusa-Huygens inequality. (English) Zbl 1466.26014

Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 115, No. 1, Paper No. 29, 12 p. (2021).
Summary: Sharp bounds of various kinds for the famous unnormalized sinc function defined by \((\sin x)/x\) are useful in mathematics, physics and engineering. In this paper, we reconsider the Cusa-Huygens inequality by solving the following problem: given real numbers \(a,b, c\in\mathbb{R}\) and \(T\in (0,\pi /2]\), we find the necessary and sufficient conditions such that the inequalities \[ \frac{\sin x}{x}>a+b\cos^cx,\quad x\in (0,T) \] and \[ \frac{\sin x}{x}<a+b\cos^cx,\quad x\in (0,T) \] hold true. In the case \(c=1\), the inequalities are extended on \((0, \pi)\). We use the elementary methods, only, improving several known results in the existing literature.

Cited in 12 Documents

MSC:

26D05 Inequalities for trigonometric functions and polynomials
26D07 Inequalities involving other types of functions
33B30 Higher logarithm functions

Keywords:

Cusa-Huygens inequality; sinc function; l’Hospital’s rule of monotonicity
Cite Review PDF
Full Text: DOI

References:

[1] Anderson, GD; Vamanamurthy, MK; Vuorinen, M., Conformal Invariants. Inequalities and Quasiconformal Maps (1997), New York: Wiley, New York · Zbl 0885.30012
[2] Bagul, YJ, Remark on the paper of Zheng Jie Sun and Ling Zhu, J. Math. Inequal., 13, 3, 801-803 (2019) · Zbl 1425.26005 · doi:10.7153/jmi-2019-13-55
[3] Bagul, Y.J., Chesneau, C.: Refined forms of Oppenheim and Cusa-Huygens type inequalities. Preprint, https://hal.archives-ouvertes.fr/hal-01972893 (2019)
[4] Bagul, YJ; Chesneau, C., Some new simple inequalities involving exponential, trigonometric and hyperbolic functions, Cubo, 21, 21-35 (2019) · Zbl 1440.26015 · doi:10.4067/S0719-06462019000100021
[5] Bagul, Y.J., Chesneau, C., Kostić, M.: The Cusa-Huygens inequality revisited. Novi Sad J. Math. (2020). (In press) · Zbl 1465.26016
[6] Bhayo, BA; Klén, R.; Sándor, J., New trigonometric and hyperbolic inequalities, Miskolc Math. Notes, 18, 1, 125-137 (2017) · Zbl 1399.26031 · doi:10.18514/MMN.2017.1560
[7] Bhayo, BA; Sándor, J., On Carlson’s and Shafer’s inequalities, Issues Anal., 3, 21, 3-15 (2014) · Zbl 1312.26030 · doi:10.15393/j3.art.2014.2441
[8] Chen, C-P; Sándor, J., Inequality chains for Wilker, Huygens and Lazarević type inequalities, J. Inequal. Appl., 8, 55-67 (2014) · Zbl 1294.26016 · doi:10.7153/jmi-08-02
[9] Chen, C-P; Cheung, W-S, Sharp Cusa and Becker-Stark inequalities, J. Inequal. Appl., 136, 2011 (2011) · Zbl 1275.26025
[10] Dhaigude, RM; Chesneau, C.; Bagul, YJ, About trigonometric-polynomial bounds of sinc function, Math. Sci. Appl. E-Notes, 8, 1, 100-104 (2020)
[11] Gradshteyn, IS; Ryzhik, IM, Table of Integrals, Series and Products (2007), Amsterdam: Elsevier, Amsterdam · Zbl 1208.65001
[12] Huygens, C.: Oeuvres completes. Soc. Hollondaise Sci. pp. 1888-1940, (1895)
[13] Malešević, B., Nenezić, M., Zhu, L., Banjac, B., Petrović, M.: Some new estimates of precision of Cusa-Huygens and Huygens approximations. Preprint arxiv:1907.00712 (2019)
[14] Mitrinović, DS, Analytic Inequalities (1970), Berlin: Springer, Berlin · Zbl 0199.38101 · doi:10.1007/978-3-642-99970-3
[15] Mortici, C., The natural approach of Wilker-Cusa-Huygens inequalities, Math. Inequal. Appl., 14, 3, 535-541 (2011) · Zbl 1222.26020
[16] Neuman, E.; Sándor, J., On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker and Huygens inequalities, Math. Inequal. Appl., 13, 4, 715-723 (2010) · Zbl 1204.26023
[17] Qi, F.; Cui, L-H; Xu, S-L, Some inequalities constructed by Tchebysheff’s integral inequality, Math. Inequal. Appl., 2, 4, 517 (1999) · Zbl 0943.26030
[18] Sándor, J., Sharp Cusa-Huygens and related inequalities, Notes Numb. Theory Discrete Math., 9, 50-54 (2013) · Zbl 1316.26020
[19] Sándor, J., Bencze, M.: On Huygen’s trigonometric inequality. RGMIA Res. Rep. Coll., 8(3), (2005)
[20] Sándor, J.; Oláh-Gal, R., On Cusa-Huygens type trigonometric and hyperbolic inequalities, Acta. Univ. Sapientiae Math., 4, 2, 145-153 (2012) · Zbl 1296.26051
[21] Zhu, L., Sharp inequalities of Mitrinović-Adamović type, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 113, 2, 957-968 (2019) · Zbl 1423.26052 · doi:10.1007/s13398-018-0521-0
[22] Zhu, L., An unity of Mitrinović-Adamović and Cusa-Huygens inequalities and the analogue for hyperbolic functions, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 113, 3-4 (2019) · Zbl 1429.26041
[23] Zhu, L., New Cusa-Huygens type inequalities., AIMS Math., 5, 5, 5320-5331 (2020) · Zbl 1484.26109 · doi:10.3934/math.2020341
[24] Zhu, L., New Mitrinović-Adamović type inequalities, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 114, 3, 119 (2020) · Zbl 1437.26017 · doi:10.1007/s13398-020-00848-w
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.